To Fox: about proofs.

就说这么多了。争吵真累，俺还是回去做俺的proof 了。

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(1) I don’t see much difference between the problems
  (A)  Find the answer to XYZ, and
  (B)  Prove that the answer to XYZ is W.
In some  sense, (B) is easier since it provides a hint.  I think it is a training process— without any goal the problem might be too hard.  In research, once you have an open problem, you make many small conjectures and tackle one at a time.  Each time you guess what might happen and try to prove (or disprove).

(2)  You said that “Problem solving encourages taking various approaches to confirm your answers. When you get the same answer from different approaches, you are more certain that the answer is correct.
 While in proofs, the correct answer is already there. So I, for instance, would stop looking for different approaches to solve the same problem once I hit the correct statement. In this sense, problem solving is better than proofs. “

— The correct final answer is always unique,  right?  The difference lies in how one gets it.  This applies to both open-end problems and proof-problems.

I don’t know why you are not interested in looking for different approaches for a proof-problem. In this forum people offer different arguments whenever a problem is posted, including proof ones.  Check a math book, how many proofs are there for the Pythagorean  theorem?  Each time a new proof is developed, it helps people understand more about the math. Maybe this is not what you are interested ?


(3)  In some cases, it is just not possible to get the result without proofs. How did people realize that not every number is rational?   How did we know that one can not trisect an angle with only straight rules and compasses?  Not every problem has a constructive solution yet—  it would be highly appreciated if you can find one. 

4) In my experience proving is an indispensable part of mathematics. One can not really distinguish proofs from constructive solutions.  Maybe what you don’t like is the easy proofs in K-12?  Well, everything has  a start.  I am happy that my daughter has a math teacher this year who emphasizes on proofs. When she tells me that “in math, you either get it or you don’t. There is nothing to memorize”, I know she is getting it.
 

所有跟帖： 

• 顶子坦数学第一人！ -海魂衫- ♂ (0 bytes) () 01/15/2015 postreply 15:25:53

• 你们俩的职业不同啊，用数学的方式也就不一样：） --百科-- ♀ (0 bytes) () 01/15/2015 postreply 15:35:36

• Agreed--I'd venture to say 95 percent of people -SwiperTheFox- ♂ (82 bytes) () 01/15/2015 postreply 15:46:30

• 这个就是你不对了，你还死守着没用到就不用学的理念。这个理念肯定在现代行不通：） --百科-- ♀ (0 bytes) () 01/15/2015 postreply 15:50:21

• I thought your work is beyond just calling for a program and -trivial- ♀ (156 bytes) () 01/15/2015 postreply 17:52:28

• 原来他还需要用数学？ -trivial- ♀ (118 bytes) () 01/15/2015 postreply 18:21:38

• 顶组长 -SheepMom2014- ♀ (0 bytes) () 01/15/2015 postreply 15:36:11

• Like it. -CirrusCloud- ♀ (0 bytes) () 01/15/2015 postreply 15:36:22

• I regard our conversation as discussion:-) -SwiperTheFox- ♂ (1666 bytes) () 01/15/2015 postreply 15:38:40

• What we see in College is that -trivial- ♀ (1075 bytes) () 01/15/2015 postreply 18:18:37

• 站你这边，再踢狐狸一脚。哈哈。 -Rock.rose- ♀ (0 bytes) () 01/15/2015 postreply 15:51:17

• 哈哈哈哈 --百科-- ♀ (0 bytes) () 01/15/2015 postreply 15:53:14

• 你离他近啊？ 那就再补一脚吧。 -trivial- ♀ (37 bytes) () 01/15/2015 postreply 18:23:29

• 楼上几位支持的，一并谢过！ 还是嫡系的好啊 ：） -trivial- ♀ (0 bytes) () 01/15/2015 postreply 18:25:31